Recall that Ampère's Law says that the circulation of the magentic field around any imaginary closed loop (not around a physical loop, such as a wire) is proportional to the current enclosed by the loop, \begin{eqnarray*} \oint\limits_{\rm loop} \BB \cdot d\rr = \mu_0 \, I_{\rm enclosed} \end{eqnarray*} But the enclosed current is just the flux integral of the current density through any surface bounded by the loop, \begin{eqnarray*} I_{\rm enclosed} = \Int_{\rm surface} \JJ\cdot d\AA \end{eqnarray*} (Why does it not matter which surface you use?)

So, now we have \begin{eqnarray*} \oint\limits_{\rm loop} \BB \cdot d\rr = \mu_0 \Int_{\rm surface} \!\! \JJ\cdot d\AA \end{eqnarray*} Stokes' Theorem tells us that the circulation integral can be replaced with the integral of the curl of the vector field over any surface bounded by the loop. We will choose the surface to be the same as the surface used to caluculate the flux of the current. \begin{eqnarray*} \Int_{\rm surface} \!\! (\grad\times\BB)\cdot d\AA = \mu_0 \Int_{\rm surface} \!\! \JJ\cdot d\AA \end{eqnarray*} Because this last relationship is true for any closed loop, we can conclude that the integrands themselves must be equal, that is, \begin{eqnarray*} \grad\times\BB = \mu_0 \,\JJ \end{eqnarray*} This is the differential form of Ampère's Law, and is one of Maxwell's Equations. It states that the curl of the magnetic field at any point is the same as the current density there. Another way of stating this law is that the current density is a source for the curl of the magnetic field.

In the activity earlier this week, Ampère's Law was used to derive the magnetic field for a symmetric current distribution. The differential form of Ampère's Law makes it possible to go the other way and find the current distribution from a given magnetic field.